On f-vectors of polytopes and matroids

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The f-vector of a simplicial complex is a fundamental invariant that counts the number of faces in each dimension. A natural question in the theory of simplicial complexes is to understand the relationship between the f-vector of the simplicial complex and the properties of the topological, algebraic and combinatorial structures associated to the simplicial complex. This dissertation studies this question for two different classes of simplicial complexes: simplicial polytopes and matroid independence complexes. For the class of simplicial polytopes we study what are the possible f-vectors of polytopes that are good approximations of a convex body with smooth enough boundary. In particular, in Chapter 2, we settle a longstanding conjecture of Kalai asserting that good approximations of smooth convex body K are far from extremal in the sense of the lower bound theorem in a precise way. We make the result quantitative in the case the convex body is of type C^2. Little is known about f-vectors of matroid independence complexes. A full characterization is believed to be out of reach and several conjectures about properties of such vectors are wide open. A famous one is a conjecture of Stanley that predicts certain behavior based on the properties of the Stanley-Reisner ring of the matroid independence complex. The main goal of the second part of this document is to study this conjecture. In Chapter 3, we prove that the conjecture holds for rank 4 matroids by means of a new combinatorial method. In Chapter 4 we study the external activity complex of a matroid which is topologically simpler than the independence complex of the matroid and contains the information of $f$-vector. Chapter 5 introduces the notion of a quasi-matroidal class of ordered simplicial complexes, the notion is used to provide extensions of various properties of matroids, including a refinement of Stanley's conjecture that is proved to hold in a variety of cases, including Schubert matroids.